THE LUSIN THEOREM AND HORIZONTAL GRAPHS
IN THE HEISENBERG GROUP
PIOTR HAJLASZ, JACOB MIRRA
Abstract. In this paper we prove that every collection of measurable functions fα , |α| = m coincides a.e. with mth order derivatives of a function g ∈ C m−1 whose derivatives of order m − 1
may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin,
Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal
a.e.
1. Introduction
In 1917 Lusin [8] proved that for every measurable function f : R →
R there is a continuous function g : R → R that is differentiable a.e.
and such that g 0 (x) = f (x) for almost all x ∈ R. This result was
generalized by Moonens and Pfeffer [9] to the case of functions defined
in Rn and then by Francos [6] to the case of higher order derivatives
in Rn . He proved that if fα , |α| = m are measurable functions in an
open set Ω ⊂ Rn , then there is a function g ∈ C m−1 (Ω) that is m times
differentiable a.e. and such that for all |α| = m, Dα g = fα a.e. It
m−1,1
is easy to see that in general one cannot require that g ∈ Cloc
, i.e.
one cannot assume that the derivatives of order m − 1 are Lipschitz
continuous. For example in the case m = 1 one cannot find a locally
Lipschitz continuous function g on R2 such that ∇g(x, y) = h2y, −2xi.
Indeed, for such a function we would have
Z 1
Z 1
∂g
∂g
(t, 0) dt +
(1, t) dt = g(0, 0) − 2,
g(1, 1) = g(0, 0) +
0 ∂y
0 ∂x
Z 1
Z 1
∂g
∂g
g(1, 1) = g(0, 0) +
(0, t) dt +
(t, 1) dt = g(0, 0) + 2
0 ∂y
0 ∂x
2000 Mathematics Subject Classification. Primary 46E35; Secondary 46E30.
Key words and phrases. Lusin theorem, Heisenberg group, characteristic points.
P.H. was supported by NSF grant DMS-1161425.
1
2
PIOTR HAJLASZ, JACOB MIRRA
which is impossible.
Clearly, continuity of derivatives of order m − 1 in Francos’ theorem
result from some uniform convergence and one could expect that with
keeping track of estimates it should be possible to prove Hölder continuity of derivatives of order m − 1. However, as we will see, a much
stronger result is true. Namely we shall prove that it is possible to
construct a function g with any modulus of continuity of derivatives of
order m − 1 which is worse than that of a Lipschitz function.
Theorem 1.1. Let Ω ⊂ Rn be open, m ≥ 1 an integer, and let f =
(fα )|α|=m , be a collection of measurable functions fα : Ω → R, |α| = m.
Let σ > 0 and let µ : [0, ∞) → [0, ∞) be a continuous function with
µ(0) = 0 and µ(t) = O(t) as t → ∞. Then there is a function g ∈
C m−1 (Rn ) that is m-times differentiable a.e., and such that
(i) Dα g = fα a.e. on Ω for all |α| = m;
(ii) kDγ gkL∞ (Rn ) < σ for all 0 ≤ |γ| ≤ m − 1;
(iii)
|Dγ g(x) − Dγ g(y)| ≤ σ|x − y|
for all x, y ∈ Rn and all 0 ≤ |γ| ≤ m − 2;
(iv)
|x − y|
|Dγ g(x) − Dγ g(y)| ≤
µ(|x − y|)
n
for all x, y ∈ R and all |γ| = m − 1.
In particular, we can take g such that the derivatives Dγ g, |γ| = m − 1
are λ-Hölder continuous simultaneously for all λ ∈ (0, 1).
Here µ(t) = O(t) as t → ∞ means that µ(t) ≤ Ct for all t ≥ t0 .
As an application of this theorem we construct horizontal graphs in
the Heisenberg group, see Theorem 3.2. For a related construction, see
also [2].
The paper is organized as follows. In Section 2 we prove Theorem 1.1. In Section 3 we provide a brief introduction to the Heisenberg
group and then we provide a construction of horizontal graphs based
on Theorem 1.1. The notation is pretty standard. By C we will denote
a general constant whose value may change within a single string of
estimates. By writing C(n, m) we mean that the constant depends on
parameters n and m only. The symbol Ccm (Ω) will stand for the class
of compactly supported C m functions.
GRAPHS IN THE HEISENBERG GROUP
3
2. Proof of Theorem 1.1
Parts (1)-(4) of the next lemma are due to Francos [6, Theorem 2.4]
and it is a generalization of an earlier result of Alberti [1, Theorem 1].
Estimates (5) and (6) are new.
Lemma 2.1. Let Ω ⊂ Rn be open with |Ω| < ∞. Let m ≥ 1 be an
integer, and let f = (fα )|α|=m , be a collection of measurable functions
fα : Ω → R, |α| = m. Let µ : [0, ∞) → [0, ∞) be a continuous function
with µ(0) = 0 and µ(t) = O(t) as t → ∞. Let ε, σ > 0. Then there is
a function g ∈ Ccm (Ω) and a compact set K ⊂ Ω such that
(1) |Ω \ K| < ε;
(2) Dα g(x) = fα (x) for all x ∈ K and |α| = m;
(3)
1
kDα gkp ≤ C(n, m)(ε/|Ω|) p −m kf kp
for all |α| = m and 1 ≤ p ≤ ∞;
(4) kDγ gk∞ < σ for all 0 ≤ |γ| < m;
(5)
|Dγ g(x) − Dγ g(y)| ≤ σ|x − y|
for all x, y ∈ Rn and all 0 ≤ |γ| ≤ m − 2;
(6)
|Dγ g(x) − Dγ g(y)| ≤
|x − y|
µ(|x − y|)
for all x, y ∈ Rn and all |γ| = m − 1.
Proof. For the proof of existence of g ∈ Ccm (Ω) with properties (1)(4), see [6, Theorem 2.4]. We need to prove that g can be modified in
such a way that (5) and (6) are also satisfied.
Let K 0 ⊂ Ω be a compact set such that |Ω \ K 0 | < ε/2 and f |K 0 is
bounded. Let f˜ = f χK 0 , where χK 0 is the characteristic function of K 0 .
Clearly kf˜k∞ < ∞. By continuity of µ we can find δ > 0 such that
µ(t) ≤ √
εm
nC(n, m)|Ω|m kf˜k∞
for all 0 ≤ t ≤ δ.
Here C(n, m) is the constant from the inequality at (3). In particular
if 0 < |x − y| ≤ δ, then
√
nC(n, m)ε−m |Ω|m kf˜k∞ ≤
1
.
µ(|x − y|)
4
PIOTR HAJLASZ, JACOB MIRRA
Let M = sup{µ(t)/t : t ≥ δ}. M is finite, because µ(t) = O(t) as
t → ∞. Applying (1)-(4) to f˜ we can find g ∈ Ccm (Rn ) and a compact
set K 00 ⊂ Ω such that
(1’) |Ω \ K 00 | < ε/2;
(2’) Dα g(x) = fα (x) for all x ∈ K 0 ∩ K 00 and |α| = m;
(3’)
1
kDα gkp ≤ C(n, m)(ε/|Ω|) p −m kf˜kp
for all |α| = m and 1 ≤ p ≤ ∞;
(4’)
σ
1
γ
kD gk∞ < min √ ,
,
n 2M
for all 0 ≤ |γ| < m.
Let K = K 0 ∩ K 00 . Then |Ω \ K| < ε and it is easy to see that the
function g has the properties (1)-(4) from the statement of the lemma.
We are left with the proof of the properties (5) and (6).
If 0 ≤ |γ| ≤ m − 2, then (4’) yields
|Dγ g(x) − Dγ g(y)| ≤ k∇Dγ gk∞ |x − y| ≤ σ|x − y|.
Let now |γ| = m − 1. If |x − y| ≥ δ, then
|Dγ g(x) − Dγ g(y)| ≤ 2kDγ gk∞ ≤
1
|x − y|
≤
.
M
µ(|x − y|)
If 0 < |x − y| < δ, then (3’) with p = ∞ yields
|Dγ g(x) − Dγ g(y)| ≤ k∇Dγ gk∞ |x − y|
√
≤
nC(n, m)ε−m |Ω|m kf˜k∞ |x − y| ≤
The proof is complete.
|x − y|
.
µ(|x − y|)
2
Now we can complete the proof of Theorem 1.1. We follow the
argument used in [6] and [9], and the only main modification is that
we are using improved estimates from Lemma 2.1.
Let U1 = Ω ∩ B(0, 1). Let V1 ⊂⊂ U1 be open with with |U1 \V1 | <
1/4. Using Lemma 2.1, we can find a compact set K1 ⊂ V1 with
|V1 \ K1 | < 1/4 and a function g1 ∈ Ccm (V1 ) such that
(a) Dα g1 (x) = fα (x) for all |α| = m and x ∈ K1 ;
(b) |Dγ g1 (x)| < 2−1 σ min {dist2 (x, U1c ) , 1}, for all x ∈ Rn and
|γ| < m;
GRAPHS IN THE HEISENBERG GROUP
5
(c)
|Dγ g1 (x) − Dγ g1 (y)| ≤ 2−1 σ|x − y|
for all x, y ∈ Rn and all 0 ≤ |γ| ≤ m − 2;
(d)
|Dγ g1 (x) − Dγ g1 (y)| ≤ 2−1
|x − y|
µ (|x − y|)
for all x, y ∈ Rn and all |γ| = m − 1.
We now proceed with an inductive definition. Suppose that the
sets K1 , . . . , Kk−1 , and the functions g1 , . . . , gk−1 are defined, for some
k ≥ 2. Let Uk = Ω ∩ B(0, k) \ (K1 ∪ . . . ∪ Kk−1 ). Let Vk ⊂⊂ Uk be open
with with |Uk \ Vk | < 2−k−1 . Using Lemma 2.1, we find a compact set
Kk ⊂ Vk with |Vk \Kk | < 2−k−1 and a function gk ∈ Ccm (Vk ) such that
P
α
(a’) Dα gk (x) = fα (x) − k−1
j=1 D gj (x), forall |α| = m and x ∈ Kk ;
(b’) |Dγ gk (x)| < 2−k σ min dist2 (x, Ukc ) , 1 , x ∈ Rn , |γ| < m;
(c’)
|Dγ gk (x) − Dγ gk (y)| < 2−k σ|x − y|
for all x, y ∈ Rn and all 0 ≤ |γ| ≤ m − 2;
(d’)
|Dγ gk (x) − Dγ gk (y)| ≤ 2−k
|x − y|
µ (|x − y|)
for all x, y ∈ Rn and all |γ| = m − 1.
P
We now take g = ∞
k=1 gk . We will prove that g satisfies claim of the
theorem. First, to see that g ∈ C m−1 (Rn ), we observe that, by (b’),
∞
X
kDγ gk kL∞ (Rn ) < σ
k=1
for all |γ| ≤ m−1, which implies C m−1 differentiability, and this proves
(ii). Properties S
(iii) and (iv) now follow immediately from (c’) and
(d’). Let C = ∞
k=1 Kk . We have |Ω \ C| = 0. We are left with
the proof that g is m-times differentiable at all points of C and that
Dα g = fα , |α| = m on C. Fix x ∈ C. Then x ∈ Kk for some k.
P
P
We write g = p + q where p = kj=1 gj and q = ∞
j=k+1 gj . Now by
α
(a’), we have D p(x) = fα (x) for |α| = m, so we are left to show that
q is m-times differentiable at x and Dα q(x) = 0 for |α| = m. Fix
|γ| = m − 1 and consider, for 0 6= h ∈ Rn , the difference quotient
6
PIOTR HAJLASZ, JACOB MIRRA
|Dγ q(x + h) − Dγ q(x)| |h|−1 . We actually have Dγ q(x) = 0, because
x ∈ Kk and supp Dγ gj ∩ Kk = ∅ for j > k. Hence
∞
|Dγ q(x + h) − Dγ q(x)|
1 X
≤
|Dγ gj (x + h)| .
|h|
|h| j=k+1
If Dγ gj (x + h) 6= 0, j ≥ k + 1, then x + h ∈ Uj . In this case, since
also x ∈ Kk ⊂ Ujc , we must have dist(x + h, Ujc ) ≤ dist(x + h, x) = |h|.
Hence by (b’) we have |Dγ gj (x + h)| ≤ 2−j σ|h|2 . Thus
∞
|Dγ q(x + h) − Dγ q(x)|
1 X −j
≤
2 σ|h|2 ≤ σ|h| → 0 as h → 0
|h|
|h| j=k+1
which proves that the derivative DDγ q(x) = 0 equals zero for any
|γ| = m − 1. This also completes the proof of (i).
In particular, if we define


0
µ(t) = | log t|−1

et
t=0
0 < t ≤ e−1
t > e−1 ,
then evidently µ satisfies the hypotheses above of the theorem, and for
every λ ∈ (0, 1) there is number Cλ > 0 such that
µ(t) > Cλ t1−λ ,
t>0
In that case derivatives Dγ g, |γ| = m − 1 satisfy
|Dγ g(x) − Dγ g(y)| ≤ Cλ−1 |x − y|λ
for all x, y ∈ Rn and λ ∈ (0, 1).
The proof is complete.
2
3. The Heisenberg group
In this section we will show how to use Theorem 1.1 to construct horizontal graphs in the Heisenberg group. While our construction works
for groups Hn , for the sake of simplicity of notation we will restrict to
the group H1 ; the generalization to the case of Hn is straightforward.
For more information about the Heisenberg group and for references to
results that are quoted here without proof, see for example [4].
The Heisenberg group is a Lie group H1 = C×R = R3 equipped with
the group law
(z, t) ∗ (z 0 , t0 ) = z + z 0 , t + t0 + 2 Im zz 0 .
GRAPHS IN THE HEISENBERG GROUP
7
A basis of left invariant vector fields is given by
(3.1)
X=
∂
∂
∂
∂
∂
+ 2y , Y =
− 2x , and T =
.
∂x
∂t
∂y
∂t
∂t
Here and in what follows we use notation (z, t) = (x, y, t). The Heisenberg group is equipped with the horizontal distribution HH1 , which is
defined at every point p ∈ H1 by
Hp H1 = span {X(p), Y (p)}.
The distribution HH1 is equipped with the left invariant metric g such
that the vectors X(p), Y (p) are orthonormal at every point p ∈ H1 .
An absolutely continuous curve γ : [a, b] → H1 is called horizontal
if γ 0 (s) ∈ Hγ(s) H1 for almost every s. The Heisenberg group H1 is
equipped with the Carnot-Carathéodory metric dcc which is defined as
the infimum of the lengths of horizontal curves connecting two given
points. The lengths of curves are computed with respect to the metric
g on HH1 . It is well known that any two points in H1 can be connected
by a horizontal curve and hence dcc is a true metric. Actually, dcc is
topologically equivalent to the Euclidean metric. Moreover, for any
compact set K there is a constant C ≥ 1 such that
(3.2)
C −1 |p − q| ≤ dcc (p, q) ≤ C|p − q|1/2
for all p, q ∈ K. In what follows H1 will be regarded as a metric
space with metric dcc . The Heisenberg group is an example of a subRiemannian manifold [7].
It is often more convenient to work the Korányi metric which is biLipschitz equivalent to the Carnot-Carathéodory metric, but is much
easier to compute. The Korányi metric is defined by
1/4
dK (p, q) = kq −1 ∗ pkK , where k(z, t)kK = |z|4 + t2
.
For nonnegative functions f and g we write f ≈ g if C −1 f ≤ g ≤ Cf
for some constant C ≥ 1. Thus bi-Lipschitz equivalence of metrics
means that dK ≈ dcc . A straightforward computation shows that for
p = (z, t) = (x, y, t) and q = (z 0 , t0 ) = (x0 , y 0 , t0 ) we have
(3.3)
dK (p, q) ≈ |z − z 0 | + |t − t0 + 2(x0 y − xy 0 )|1/2 .
The inequality (3.2) implies that the identity mapping from H1 to
R is locally Lipschitz, but its inverse is only locally Hölder continuous
with exponent 1/2. One can prove that the Hausdorff dimension of any
open set in H1 equals 4 and hence H1 is not bi-Lipschitz homeomorphic
to R3 , not even locally.
3
8
PIOTR HAJLASZ, JACOB MIRRA
Gromov [7, 0.5.C] posted the following question: Given two subRiemannian manifolds V and W and 0 < α ≤ 1, describe the space
of C α maps f : V → W . He also asked explicitly about the existence
of Hölder continuous embeddings and homeomorphisms. In particular
Gromov [7, Corollary 3.1.A], proved that if f : R2 → H1 is a C α embedding, then α ≤ 2/3 and he conjectured that α = 1/2. This
leads to a search for various surfaces in the Heisenberg group with
interesting geometric properties from the perspective of the CarnotCarathéodory metric and with suitable estimates for the Hölder continuity of a parametrization.
A problem which is related, but of independent interest, is that of
finding estimates for the size of the characteristic set on a surface S
in H1 . We say that a point on a surface in the Heisenberg group is
characteristic if the tangent plane at this point is horizontal. The
characteristic set C(S) is the collection of all characteristic points on
S. In general the Hausdorff dimension of C(S) on a regular surface
s
is small. Denote by HE
and dimE the Hausdorff measure and the
Hausdorff dimension with respect to the Euclidean metric. Balogh [2]
proved that if S is a C 2 surface in H1 , then dimE (C(S)) ≤ 1 and if S
1,1
is a C
then dimE C(S) < 2. On the other hand he proved
T surface,
2
that 0<α<1 C 1,α surfaces may satisfy HE
(C(S)) > 0. For other related
results, see [5]. We should also mention the paper [3] that contains a
construction of horizontal fractals being graphs of BV functions.
These questions motivated us in the construction of the example
that we describe next (Theorem 3.2). In what follows we will investigate surfaces in H1 being graphs of continuous functions of variables (x, y). Given a function u : Ω → R, Ω ⊂ R2 we denote by
Φ(x, y) = (x, y, u(x, y)) the canonical parametrization of the graph.
We regard Φ as a mapping from Ω to H1 .
Proposition 3.1. Suppose Ω ⊂ R2 is bounded. Then u is α-Hölder
continuous, α ∈ (0, 1] if and only if Φ : Ω → H1 is α/2-Hölder continuous.
Proof. Suppose that u is α-Hölder continuous. We need to prove
that
dK (Φ(z), Φ(z 0 )) ≈ |z − z 0 | + |u(z) − u(z 0 ) + 2(x0 y − xy 0 )|1/2
(3.4)
≤ C|z − z 0 |α/2 .
GRAPHS IN THE HEISENBERG GROUP
9
Since Ω is bounded and t ≤ Ctα/2 for 0 ≤ t ≤ t0 , we have |z − z 0 | ≤
C|z − z 0 |α/2 for all z, z 0 ∈ Ω. Similarly boundedness of Ω yields
√
|2(x0 y − xy 0 )|1/2 ≤
2(|y| |x − x0 | + |x| |y − y 0 |)1/2 ≤ C|z − z 0 |1/2
(3.5)
≤ C 0 |z − z 0 |α/2 .
The above estimates and the α-Hölder continuity of u readily imply
(3.4).
Suppose now that Φ is α/2-Hölder continuous i.e., (3.4) is true. The
triangle inequality, (3.5) and (3.4) yield
|u(z) − u(z 0 )|1/2 ≤ |u(z) − u(z 0 ) + 2(x0 y − xy 0 )|1/2 + C|z − z 0 |α/2
≤ C 0 |z − z 0 |α/2
which in turn implies α-Hölder continuity of u. The proof is complete.
2
If u : Ω → R, Ω ⊂ R2 is Lipschitz continuous, then u is differentiable
a.e. and hence the graph of u has the tangent plane for a.e. z ∈
Ω. However, it cannot happen that the tangent plane to the graph
is horizontal a.e. Indeed, it is well known and easy to check that the
tangent plane at (x, y, u(x, y)) is horizontal if and only if
∂u
∂u
= 2y and
= −2x,
∂x
∂y
but we have already checked at the beginning of this article that this
system of equations admits no Lipschitz solutions. However, we have
the following result.
Theorem 3.2. Let µ : [0, ∞) → [0, ∞) be a continuous function such
that µ(0) = 0 and µ(t) = O(t) as t → ∞. Then there is a continuous
function u : R2 → R such that
(1) |u(x) − u(y)| ≤ |x − y|/µ(|x − y|) for all x, y ∈ R2 ;
(2) u is differentiable a.e.;
(3) the tangent plane to the graph of u is horizontal for almost all
(x, y) ∈ R2 .
This result is a straightforward consequence of Theorem 1.1 in the
case of m = 1 and f1 = 2y, f2 = −2x.
Note that for almost all (x, y) ∈ Rn the corresponding points on
the surface are characteristic. The result is sharp – any modulus of
continuity stronger than that in (1) would mean that the function is
Lipschitz continuous and for such functions there are no surfaces with
10
PIOTR HAJLASZ, JACOB MIRRA
the property (3). Proposition 3.1 allows one to reinterpret the theorem
in terms of Hölder continuous surfaces with horizontal tangent planes.
We leave details to the reader.
References
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110–118.
[2] Balogh, Z. M.: Size of characteristic sets and functions with prescribed
gradient. J. Reine Angew. Math. 564 (2003), 63–83.
[3] Balogh, Z. M., Hoefer-Isenegger, R., Tyson, J. T.: Lifts of Lipschitz
maps and horizontal fractals in the Heisenberg group. Ergodic Theory Dynam.
Systems 26 (2006), 621–651.
[4] Capogna, L., Danielli, D., Pauls, S. D., Tyson, J. T.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem.
Progress in Mathematics, 259. Birkhäuser Verlag, Basel, 2007.
[5] Franchi, B., Wheeden, R.L.: Compensation couples and isoperimetric
estimates for vector fields. Colloq. Math. 74 (1997), 9–27.
[6] Francos, G.: The Luzin theorem for higher-order derivatives. Michigan
Math. J. 61 (2012), 507–516.
[7] Gromov, M.: Carnot-Carathéodory spaces seen from within. SubRiemannian geometry, pp. 79–323, Progr. Math., 144, Birkhäuser, Basel,
1996.
[8] Lusin, N.: Sur la notion de l’integrale. Ann. Mat. Pura Appl. 26 (1917),
77-129.
[9] Moonens, L., Pfeffer, W. F.: The multidimensional Luzin theorem. J.
Math. Anal. Appl. 339 (2008), 746–752.
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA, [email protected]
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA, [email protected]